Question

The measures of two angles of a parallelogram differ by 52 degrees. The number of degrees in the smaller angle is

• A. 38
• B. 52
• C. 64
• D. 76
• E. 128

Hint:

Remembering the properties of shapes is key in geometry. For parallelograms, the relationship between consecutive angles is a very common topic for questions.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem uses the properties of angles in a parallelogram. Key properties are: (1) Opposite angles are equal. (2) Consecutive angles (angles next to each other) are supplementary, meaning they add up to 180 degrees. Since the two angles differ, they must be consecutive angles.
Step 2: Key Formula or Approach:
We can set up a system of two linear equations. Let the two consecutive angles be \(x\) and \(y\).
Equation 1 (Supplementary): \(x + y = 180\)
Equation 2 (Difference): \(x - y = 52\) (assuming \(x\) is the larger angle)
Step 3: Detailed Explanation:
We have the system:
1) \(x + y = 180\)
2) \(x - y = 52\)
We can solve this system by elimination. Add the two equations together:
\[ (x + y) + (x - y) = 180 + 52 \]
\[ 2x = 232 \]
\[ x = \frac{232}{2} = 116 \]
So the larger angle is 116 degrees.
Now substitute the value of \(x\) back into the first equation to find \(y\):
\[ 116 + y = 180 \]
\[ y = 180 - 116 = 64 \]
The smaller angle is 64 degrees. The two angles are 116\textsuperscript{o} and 64\textsuperscript{o}. Their difference is \(116 - 64 = 52\), and their sum is \(116 + 64 = 180\). The calculations are correct.
Step 4: Final Answer:
The number of degrees in the smaller angle is 64.